Knife Edges And Pivots

• Introduction
• The Model
• The Selection
• Postscript
• Further reading
• Related case studies

Introduction

Middle-aged readers may remember expensive watches with the words "17 Sapphires" printed on the face, roughly where the word "Quartz" now appears. The message was that the mechanism had bearings made of sapphire (Al₂O₃) and that that was good. A really expensive watch had, not sapphires, but diamonds.

They are examples of good materials for knife edges and pivots. These are bearings in which two members are loaded together in nominal line or point contact, and can tilt relative to one another, or rotate freely about the load axis (Figure 13.1). The main requirements of materials for such bearings are high hardness (to carry the contact pressures) and high modulus (to give positional precision and to minimize frictional losses). But in what combination? The answer depends on the objective: maximum precision or maximum load-capacity. Table 13.1 summarizes the requirements.

Figure 13.1 A knife edge and a pivot. They have small contact area, for precision, and high strength, to avoid damage by contact stresses.

Table 13.1 The design requirements

The Model

Consider maximizing precision first. The positional accuracy of a pivot or knife edge increases, and its frictional losses decrease as the area of contact A is made smaller and smaller. To maximize precision, we seek to minimize A, but as we do so, the contact pressure rises. For a given load, F, the peak contact pressure p is given by Herzian contact theory; it is:

(M13.1)

If both surfaces are to remain elastic, this pressure must not exceed their hardness, H, divided by a safety factor, which we ignore. (It does not affect the outcome of the analysis.) Thus

(M13.2)

Thus the area of contact is minimized, and the precision maximized, by selecting materials with large values of the performance index

(M13.3)

Now consider the alternative objective: that of maximizing the load that the pivot can support at fixed geometry. Herzian contact theory gives the maximum pressure in the contact zone of a hemispherical pivot-tip pressed against a flat block by a force F as

(M13.4)

where R is the radius of curvature of the tip of the pivot, E is its elastic modulus and C is a constant close to unity. The constraint remains the same: this pressure must not exceed the hardness H of the pivot or the block. Thus

(M13.5)

The load is maximized, for a given geometry R, by choosing materials with large values of the performance index

(M13.6)

The Selection

The performance indices involve H and E. Figure 13.2 shows the appropriate chart. For precision you want high M₁, and then ceramics are definitely the best choice: Al₂O₃ (sapphire), silicon carbide and nitride, and — above all — diamond. Cermets (composites of tungsten carbide and cobalt) are the only other class of solid which competes with them. For load-bearing capacity, you want high M₂ instead; then high-carbon steels, tool steels and high-strength titanium alloys become possibilities, although ceramics and cermets remain the most attractive candidates.

Figure 13.2(a) Chart of hardness H against modulus E, showing the index M₁.

Figure 13.2(b) Chart of hardness H against modulus E, showing the index M₂.

But there is more to it than that. Change of temperature will influence the positional accuracy of a pivot. Precision is maintained by choosing materials with low thermal expansion. Robustness, on the other hand, requires that the pivot or knife-edge survive rough handling — and for this a degree of toughness is desirable. Figure 13.3 shows a chart of fracture toughness plotted against thermal expansion coefficient. Using this as a second stage, gives the selections in Tables 13.2(a) and (b).

Figure 13.3 A chart of fracture toughness, K_1c, against thermal expansion coefficient, λ, showing the selection stage for robust, precise, pivots.

MATERIAL			COMMENT
	(GPa)	(MPa)
Diamond	8000	0.2	Outstanding on all counts except cost.
Tungsten Carbide, WC	4000	0.1	Excellent M1 and M2, good corrosion resistance, poor K1c
Silicon Carbide, SiC	3000	0.1	Excellent M1 and M2, good corrosion resistance, poor K1c
Silicon Nitride, Si3N4	2200	0.11	Excellent M1 and M2, good corrosion resistance, poor K1c
Zirconium Carbide, ZrC			Excellent M1 and M2, good corrosion resistance, poor K1c
Sapphire, Al2O3	2000	0.09	Good M1 and M2, good corrosion resistance, poor K1c
Glass and Silica	1000	0.05	Good M1 but brittle — poor K1c
High-Carbon Steel	400	0.02	Good M1 and M2, poor corrosion resistance, but excellent K1c

Table 13.2(a) Materials for precise knife edges and pivots

MATERIAL			COMMENT
	(GPa)	(MPa)
High-Carbon Steel	400	0.02	Good M1 and M2, poor corrosion resistance, but excellent K1c
Cermets	500	0.03	Better than steel on all counts except cost
Silicon Carbide, SiC	3000	0.1	Excellent M1 and M2, good corrosion resistance, poor K1c
Silicon Nitride, Si3N4	2200	0.11	Excellent M1 and M2, good corrosion resistance, poor K1c
Diamond	8000	0.2	Outstanding on all counts except cost and toughness

Table 13.2(b) Materials for robust knife edges and pivots

Postscript

The final choice depends on the details of its application. In sensitive force balances and other measuring equipment, very low friction is important: then we need the exceptionally high modulus of sapphire or diamond. In high load-capacity devices (weighbridges, mechanical testing equipment), some ability to absorb overloads by limited plasticity is an advantage, and hardened steel is a good choice. If the environment is a potentially corrosive one — and this includes ordinary damp air — glass or a ceramic may be best.

Further Reading

'Tribology Handbook' (1973) Ed. Neale, MJ, Butterworths, London, Section C6.

'Elsevier Materials Selector' (1991) Eds. Waterman, NA and Ashby, MF, Elsevier, Amsterdam, Section 1.5.3, p. 358 et seq.

Related Case Studies

1. Materials for Springs

2. Elastic Hinges And Couplings

3. Materials for Static Seals

4. Diaphragms for Pressure Actuators

5. Minimising Distortion in Precision Devices